Proof of theorem 7.7.1. in Lars Hormander's "The Analysis of Linear Partial Differential Operators I" I have been trying to understand this proof for a while but I just can't follow what the author writes. If anyone can either explain Hormander's proof or has an alternate proof I would highly appreciate it.

THEOREM 7.7.1. Let $K\subset\mathbb{R}^n$ be a compact set, $X$ an open neighborhood containing $K$ and $j, k$ non-negative integers. If $u\in C^{k}_c(K), f\in C^{k+1}(X)$ and $\Im[f]\geq 0$ in $X$ then for all $w > 0$,
  $$
w^{j+k}\left\lvert \int u(x)\left(\Im[f(x)]\right)^j e^{iwi(x)} dx \right\rvert
\leq C \sum_{\lvert\alpha\rvert \leq k}\sup\lvert D^\alpha u\rvert \left(\lvert f^\prime\rvert^2 + \Im[f]\right)^{\lvert{\alpha}/2-k}
$$

Proof available here
In the statement of the theorem the author also mentions that "$C$ is bounded when $f$ stays in a bounded set in $C^{k+1}(X)$". I am not sure what he means by that. What exactly does the constant $C$ depend on?
The proof makes sense to me until equation (7.7.3). Note that the supremum is outside of the sum, so how did the author use the induction hypothesis?
I also don't see how equation (7.7.4) follows from the lemma. 
On the top of page 218, how is the inequality
$$
N\lvert{u_\nu}\rvert_\mu \leq C\left(\lvert{N}_1\lvert{u_\nu}\rvert_{\mu-1}\dots \lvert{u_\nu}\rvert_0\right)
$$
obtained?
Finally, how does (7.7.6) follow from (7.7.5)?
 A: 
First question:

I have read a bite but patently $C\equiv C(f)$ in the theorem is a generic constant depending on the data such as  function $f,$ and $g$.......
the terminology generic means  that the constant can be different from one estimate the another but still it is relabelled with  the same notation . Here the notation is $C$.  This syntax it quite useful to people dealing with a lot of estimates  it helps to simplify the understanding  of some hard frame work. I meet this several time in the context of Harmonic analysis. 
This is opposed to people dealing with optimal theory there they try while tracking optimal to use at most as possible different notation for constant for each estimate.    

Now to my experience and the for context of your theorem, 
  the statement "$C$ stays bounded as $f\in C^{k+1}(X)$" means that the obtained constants from all estimates will be such that (or satisfies )
  $$C\equiv  \sup\{C(f): f\in C^{k+1}(X)\}<\infty$$
  whenever such $f$ it concern with the estimates. I hope this helps you to finish your proof. 


Second question;
For the estimate $7.7.4$ just use translation as follows $$g(w)= Im(f(z+w))$$
then applies Lemma again with, so that for fixed $z$  you have,
  $$g(0)= Im(f(z))~~~~and~~~g'(0)= \partial_v  Im(f(z))$$ 

Addendum: Note that for small values of $k$ the problem has been solved at the very beginning of the proof. please the beginning of the proof so here $k$ it suppose to be large enough. and hence $N$ is at least twice differentiable. 


Third Question:
Last For the estimate $7.7.6$ is exactly a consequence of the general Leibniz formula  and $7.7.4$ and $7.7.5$ in fact we have , 
$$\partial_\alpha(fg) = \sum_{\gamma+\beta =\alpha} {\gamma \choose \alpha} \partial_\gamma( f)\partial_\beta(g)~~~~~~~~~~(\text{general Leibniz formula})$$
On the other hand, $$|u|_{\mu} =\sup_{|\alpha|=\mu}\{ |\partial_\alpha u|\}$$


With this all your estimates follow. 

Addendum: in the proof of 7.7.5 it says we shall now proof that when $\mu\le k$ we have 
  $$ N^{1/2}\sum_{\nu =1}^{n}|u_\nu|_\mu + N|u_0|_\mu\le C \sum_{r=1}^{\mu}|u|_r N^{(r-\mu)/2}.\tag{7.7.5}$$ the whole proof of this inequality is provided below and before 7.7.6. 

